With these preliminaries, we can give a geometric characterization
for the trajectory-based approach to discrete-time flatness suggested
in Section REF . In accordance with
the literature on static and dynamic feedback linearization for discrete-time
systems, we consider a suitable neighborhood of an equilibrium \(x_{0}=f(x_{0},u_{0})\) ,
see e.g. [1]} or [2]}.
However, we want to emphasize that for many systems the concept may
be useful even if the conditions fail to hold at an equilibrium.

The system (REF ) is said to be
flat around an equilibrium \((x_{0},u_{0})\) , if the \(n+m\)  coordinate
functions \(x\)  and \(u\)  can be expressed locally by an \(m\) -tuple
of functions
\(y^{j}=\varphi ^{j}(\zeta _{[-q_{1}]},\dots ,\zeta _{[-1]},x,u,\dots ,u_{[q_{2}]})\,,\) 
